Peer Health Shocks and Labor Supply

Guest Lecture for ECON 3100

Joshua C. Martin
joshua.martin@vanderbilt.edu

Vanderbilt University

October 15, 2024

Worker Productivity in Sports

  1. What is the one-word term associated with the usage of “sabermetrics” for identifying undervalued workers in sports settings?
    • Hint: This term originates from the title of the 2003 biographical novel of a (sports) General Manager and was later adapted into a 2011 Academy Award-nominated film.
    • What are the names of the two lead actors staring in the aforementioned film?
  1. What sport was this approach initially most famously associated with?
    • “Sabermetrics” was originally written as “SABRmetrics”. What does “SABR” stand for?

Musical Interlude

Worker Productivity in Sports

  1. What is the name of the individual credited with creation of “Moneyball”?
    • This individual graduated with degrees in what two subjects?
  1. What commonly-used concept from Labor Economics most closely approximates workers’ “Approximate Value”?

Musical Interlude

Worker Productivity in Sports

  1. If team sizes are fixed, what margin are we referring to with respect to \(MP_L\)?
  1. Thinking like an economist, list five distinct factors changing individual productivity in this setting.
  • Going forward, let worker productivity be represented by \(\upsilon\)
    • Constructed such that the value of a replacement worker is equal to zero
  1. Draw the relationship between \(\upsilon\) and age

Worker Productivity in Sports

Productivity and Age (Q. 5, 6 & 7)

Worker Productivity in Sports

  1. \(\dfrac{\delta (Salary)}{\delta \upsilon} \gtrless 0\)?
  1. Salary cap (\(\approx\) mean revenue) per worker: hockey (\(h\)) = $3.82M; football (\(f\)) $6.21M
    • \(\dfrac{\delta (Salary_f)}{\delta \upsilon_f} \gtrless \dfrac{\delta (Salary_h)}{\delta \upsilon_h}\)?
    • \(Mean(Salary_f) \gtrless Mean(Salary_h)\)
    • \(Median(Salary_f) \gtrless Median(Salary_h)\)

Worker Productivity in Sports

  1. Thinking like an economist, define “superstar” effects.
  1. In which sport are “superstar” effects more likely to appear: hockey or football?
    • Hint: Hockey has 6 players on the ice per team. Rinks are 200 \(\times\) 85ft. Football has 11 players per team. Fields are 360 \(\times\) 160ft.

Musical Interlude

Worker Productivity in Sports

Productivity and Salary (Q. 8, 9a & 11)

Worker Productivity in Sports

Salary Distribution (9b & 9c)

Worker Productivity in Sports

Distribution of Productivity by Sport (Q. 11 hint)

Health Concerns in Sports

  1. Which specific injury type is of greatest concern to workers in violent collision sports?
    • Are fears about this injury type becoming more or less salient over time?
    • Hint: what disease has recently (\(\leq\) 20 years) become synonymous with affecting these athletes?
  1. Define, using economic theory, two ways these health risks affect wages.
    • Do these ways increase or decrease wages? Is the prediction theoretically ambiguous?

Musical Interlude

Health Concerns in Sports

Timeline of CTE Diagnoses (Q. 12)

What is CTE?

Health Concerns in Sports

Theoretical Framework

Wokrers’ Maximize Utility

\[U_i = \sum_{t=T_0}^{T} \delta^{t-T_0} \left( d_{ist} \cdot \omega_{ist} + (1 - d_{ist}) \cdot \eta_{ist} \right)\]

  • \(U_i\) = utility of individual \(i\) over the periods from \(t\) to \(T\)

  • \(\delta\) = discount factor; reflects time preferences; \(0 < \delta < 1\),

  • \(d_{ist} \in \{0, 1\}\); whether employed in sports

  • \(\omega_{ist}\) = sport wage

  • \(\eta_{ist}\) = non-sport wage

Theoretical Framework

Non-Sport Wages

\[\eta_{ist} = \mu_i - \theta_{ist}\]

  • \(\mu\): worker’s human capital value

  • \(\theta\): reduction in working years due to physical toll from violent sports

\[\theta_{ist} = \gamma_{st} + \epsilon_{ist}\] - \(\gamma\) = known (“inherent”) dangers of the sport

  • Full-information of workplace safety \(\implies E[\epsilon] = 0\)

Theoretical Framework

Sport Wages

\[\omega_{ist} = \rho_{st} \cdot \upsilon_{ist} + \beta_{ist} \cdot \theta_{ist}\]

  • \(\rho\) = market demand

  • \(\upsilon\) = value of the marginal product of labor (relative to replacement worker)

  • \(\beta\) = workers’ bargaining power

\[\beta_{ist} = \mu_i + \pi_{st} \cdot \upsilon_{ist}\] - \(\pi\) = strength of collective bargaining agreement

Theoretical Framework

Predictions: Full Information

  • Workers will remain working in sports if \(\omega_{ist} \geq \eta_{ist}\)
  • Changes in the decision to remain in sports:

\[ \Delta_{it} (\upsilon_{ist} + \epsilon_{ist} \cdot (\upsilon_{ist} + 1)) \geq 0 \]

  • If \(E[\epsilon]=0\), then remain so long as productivity increases:

\[\Delta_{it}( \upsilon_{ist}) \geq 0\]

Change Operator

Theoretical Framework

Predictions: Imperfect Information

  • Workers will remain working in sports if \(\omega_{ist} \geq \eta_{ist}\)

\[ \Delta_{it} (\upsilon_{ist} + \epsilon_{ist} \cdot (\upsilon_{ist} + 1)) \geq 0 \]

  • If \(E[\epsilon] \neq 0\), then remaining depends upon changes in productivity (\(\Delta p \lessgtr -1\))
    • Those with declining productivity are more likely to exit; cannot internalize risk into wages
    • Those with rising productivity are more likely to stay; internalize risk into wages (aided by exit of colleagues with declining productivity)

Findings

  • Athletes with a former teammate who died of CTE are 2.2-4.0 percentage points (8-23%) more likely to retire.
    • Mechanisms: familiarity, recency, opportunity costs, wage flexibility
    • Placebo: no effect for non-safety-related deaths
    • Treated athletes exhibit higher risk sensitivity, opting for larger signing bonuses and shorter contracts at lower salaries.
  • “Treated” workers value this health shock disamenity at $6M and need a $1M raise to remain indifferent to exiting.

Results

Peer Health Shocks & Retirement: Football

Results

Peer Health Shocks & Retirement: Hockey

Results

Event Studies (by Sport)

Hypothesis I

Familiarity Matters

  • The modal career length for hockey is over 30% longer than for football while team sizes are over 100% smaller
    • This means that the modal hockey player is more likely to been teammates with someone who dies of CTE and will have had greater exposure to them prior to their death

Results

Peer Health Shocks & Retirement (by Time as Teammates)

Dependent Variable: \(\mathbb{1}\)(Exit) * 100
Model: (1) (2) (3)
Panel A (Time as Teammates)
CTE (Post) x Teammate 2.53*** 4.30*** 4.67***
(0.80) (1.22) (1.69)
Pre-Treatment Mean 29.03 25.69 24.87
% Change 0.09 0.17 0.19
Observations 229,833 218,853 215,226
Mean Pre-Treatment Age 27.75 28.59 29.31
Length of Time Teammates (\(L\)): \(L \geq 1\) \(L \geq 2\) \(L \geq 3\)

Treatment Effects by Age

Hypothesis II

Recency of Treatment Matters

Results

Peer Health Shocks & Retirement (Recency of Treatment)

Dependent Variable: \(\mathbb{1}\)(Exit) * 100
Model: (1) (2) (3)
Panel A (Recency of Treatment)
CTE (Post) x Teammate 3.14** 1.78* 1.91**
(1.45) (0.96) (0.88)
Pre-Treatment Mean 10.45 22.27 28.29
% Change 0.3 0.08 0.07
Observations 95,049 202,824 219,366
Recency of Treatment (D): \(D \leq 2\) \(D \leq 4\) \(D \leq 6\)

Results

Peer Health Shocks & Retirements (Recency & 3+ Years Teammates)

Dependent Variable: \(\mathbb{1}\)(Exit) * 100
Model: (1) (2) (3)
Panel B (Recency & 3+ Years Teammates)
CTE (Post) x Teammate 7.22* 4.62* 3.81*
(4.05) (2.65) (2.29)
Pre-Treatment Mean 10.45 22.27 28.29
% Change 0.69 0.21 0.13
Observations 39,231 68,256 73,035
Recency of Treatment (D): \(D \leq 2\) \(D \leq 4\) \(D \leq 6\)

Results

Peer Deaths & Retirement: A Placebo Test

Dependent Variable: \(\mathbb{1}\)(Exit) * 100
Model: (1) (2)
Car Accident (Post) x Teammate -0.01
(0.01)
Disease (Post) x Teammate 0.00
(0.01)
Mean of Dep. 0.19 0.19
% Change -0.06 0.00
Observations 95,533 70,891

Hypothesis III

Bargaining Power & Wage Flexibility Matter

  • The collective bargaining agreement covering hockey players fully guarantees salaries which comes at the expense of:
    • highly restrictive salary caps (limits on the sum of labor costs)
    • escrow payments (withheld salary; dependent upon league revenues)
  • Salary cap per worker in 2024:
    • NFL \(\approx \$6.21M\) per worker
    • NHL \(\approx \$3.82M\) per worker

Contract Data

Name that…

Contract Data

Results

Peer Health Shocks & Retirement (by Opportunity Cost)

Dependent Variable: \(\mathbb{1}\)(Exit) * 100
Sample: Full Hockey Football
Model: (1) (2) (3) (4)
CTE (Post) x Teammate 3.59** 2.48* 1.53 4.08**
(1.45) (1.46) (1.41) (1.88)
\(sinh^{-1}\)(Opp. Cost) -0.48*** -0.06*** -0.09*** -0.02
(0.01) (0.01) (0.01) (0.02)
CTE (Post) x Teammate x \(sinh^{-1}\)(Opp. Cost) -0.37*** -0.32*** -0.06 -0.52***
(0.09) (0.09) (0.08) (0.12)
Controls: Ct. Total & Length \(\checkmark\) \(\checkmark\) \(\checkmark\)
Observations 131,424 131,424 18,679 112,745

Results

Back-of-the-Envelope Calculation: Compensation and Indifference

Conclusion

  • I find that workers labor supply is highly elastic with respect to peer health shocks, even in environments where these risks are widely known and highly discussed.
    • Familiarity, recency and wage flexibility help explain this finding.
  • These findings indicate that individuals significantly underestimate the utility loss associated with risky behaviors even in settings where such risks are widely known and discussed.
    • Suggests that public safety policies can provide complementary benefits.

Thank you!

joshua.martin@vanderbilt.edu

joshmartinecon.github.io

“Let us suppose that …[a great, distant empire], with all its myriads of inhabitants, was suddenly swallowed up by an earthquake, and let us consider how a man of humanity […], who had no sort of connection with that part of the world, would be affected upon receiving intelligence of this dreadful calamity. He would, I imagine, first of all, express very strongly his sorrow for the misfortune of that unhappy people, he would make many melancholy reflections upon the precariousness of human life, and the vanity of all the labours of man, which could thus be annihilated in a moment…

And when all this fine philosophy was over, when all these humane sentiments had been once fairly expressed, he would pursue his business or his pleasure, take his repose or his diversion, with the same ease and tranquillity, as if no such accident had happened. The most frivolous disaster which could befall himself would occasion a more real disturbance.

If he was to lose his little finger to-morrow, he would not sleep to-night; but, provided he never saw them, he will snore with the most profound security over the ruin of a hundred millions of his brethren, and the destruction of that immense multitude seems plainly an object less interesting to him, than this paltry misfortune of his own.”

- Adam Smith, The Theory of Moral Sentiments

Appendix

Change Operator

  • Unit fixed-effects: \(\Delta X_i = X_{it} - \frac{1}{T} \sum_{\tau = 1}^T x_{i \tau}\)

  • Time fixed-effects: \(\Delta X_i = X_{it} - \frac{1}{I} \sum_{\iota = 1}^I x_{\iota t}\)

  • For notational simplicity, I denote unit and time fixed-effects as \(\Delta_i\) and \(\Delta_t\) respectively.

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Appendix

CTE

  • CTE is an irreversible neurodegenerative condition that arises from repeated head impacts over extended periods.
    • Symptoms: motor impairments, difficulties regulating emotion, nerve pain, and dementia, often presenting as having advanced Parkinson’s and Alzheimer’s disease.
    • Treatment: symptom management
    • Age profile: 20s - 80s
    • Diagnosis: post-mortem; analysis of brain tissue

Appendix

CTE

  • CTE has become synonymous with violent collision sports, especially American football.
  • Before the term “CTE,” post-concussive symptoms in athletes were referred to as having one’s “bell rung,” being “punch drunk,” or “shell shocked.”
    • These terms implied rare, worst-case outcomes experienced by reckless individuals, with transitory effects.

Appendix

CTE

  • All athletes diagnosed with CTE from 2005-2008 shared numerous common traits: long professional careers, disproportionate exposure to repeated head trauma, and violent styles of play.

  • This changed in 2009 with the discovery of CTE in Chris Henry.

    • The collision-averse Henry had played competitive football for fewer than ten years before he died at age 26.

  • In 14 months in 2010-2011, 4 hockey players (3 of whom were still active professionals) died and were diagnosed with CTE at the ages of 45, 35, 28, and 27. Back

Appendix

Identifying Variation (Football, Part I)

Player Career Date of Death Teammates at τ = -4
Justin Strzelczyk 1990-1998 9/30/2004 61
Shane Dronett 1992-2001 1/21/2009 40
Chris Henry 2005-2009 12/17/2009 137
Junior Seau 1990-2012 5/2/2012 157
Jovan Belcher 2009-2012 12/1/2012 116
Paul Oliver 2008-2011 9/24/2013 115
Adrian Robinson 2012-2013 5/16/2015 115

Appendix

Identifying Variation (Football, Part II)

Player Career Date of Death Teammates at τ = -4
Tyler Sash 2011-2012 9/8/2015 80
Aaron Hernandez 2010-2012 4/19/2017 71
Daniel Te’o-Nesheim 2010-2013 10/29/2017 117
George Atkinson 2014-2016 12/2/2019 117
Max Tuerk 2017-2017 6/20/2020 68
Vincent Jackson 2005-2016 2/15/2021 92
Phillip Adams 2010-2015 4/8/2021 114
Demaryius Thomas 2010-2019 12/9/2021 199

Appendix

Identifying Variation (Hockey)

Player Career Date of Death Teammates at τ = -4
Bob Probert 1986-2002 7/5/2010 53
Derek Boogaard 2006-2011 5/13/2011 102
Rick Rypien 2006-2011 8/15/2011 92
Wade Belak 1997-2011 8/31/2011 151
Steve Montador 2002-2012 2/15/2015 141
Marek Svatoš 2004-2011 11/4/2016 59

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Appendix

Peer Health Shocks & Retirement (by Age)

Dependent Variable: \(\mathbb{1}\)(Exit) * 100
Model: (1) (2) (3)
Panel B (Age)
CTE (Post) x Teammate 0.10 3.70** 0.56
(2.22) (1.53) (1.72)
Pre-Treatment Mean 10.8 17.31 36.27
% Change 0.01 0.21 0.02
Observations 97,794 131,022 32,643
Pre-Treatment Mean Age 22.36 26.03 32.25
Age Percentile (\(P\)): \(P < \frac{1}{3}\) \(\frac{1}{3} \leq P < \frac{2}{3}\) \(P \geq \frac{2}{3}\)

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References

Omalu, Bennet I., Steven T. DeKosky, Ryan L. Minster, M. Ilyas Kamboh, Ronald L. Hamilton, and Cyril H. Wecht. 2005. “Chronic Traumatic Encephalopathy in a National Football League Player.” Neurosurgery 57 (1): 128–34. https://doi.org/10.1227/01.neu.0000163407.92769.ed.
Omalu, Bennet I, Steven T DeKosky, Ronald L Hamilton, Ryan L Minster, M Ilyas Kamboh, Abdulrezak M Shakir, and Cyril H Wecht. 2006. “Chronic Traumatic Encephalopathy in a National Football League Player: Part II.” Neurosurgery 59 (5): 1086–93.